I am working on this exercise and I wonder if it is necessary that X and Y are also complete:
Show that if $(X,||\cdot||)$ and $(Y,||\cdot||)$ are normed vector spaces, then the product space $X \times Y$ is a normed vector space if we define:
$||(x,y)||=max(||x||,||y||)$.
If they are also complete, then they are Banach spaces and the answer is this link: Given $X,Y$ are Banach spaces with norms $\|x\|_X,\|y\|_Y$, prove $\|(x,y)\|=\max(\|x\|_X,\|y\|_Y)$ is a norm and defines a Banach space
Could you please help me?
You do not need completeness. $X\times Y$ has a natural vector space structure (usually it is written $X\oplus Y$). Then you may show that the max norm satisfies the four requirements of norms: nonnegativity, linear homogeneity, the triangular inequality and definiteness – all of which has been done within the question you’ve quoted.
Completeness is a totally separate condition. As you can see from that question, it wasn’t used anywhere in the first part of the proof.